I am learning about differentiable manifold, and I got a question asking to give a reason why I need at least two charts to cover a compact manifold. I know there is a question here also related to this, I got some idea from that question but also generated one more confusions.
My idea is:
If $M$ is a compact manifold covered by a single chart $(U,\phi)$, then by the definition (the locally Euclidean), $U$ is homeomorphic to the open subset $\phi(U) \subset R^{n}$ for some $n$.
But the only connected open compact subset of $R^{n}$ is empty set for $n \geq 1$, if $n=0$ we also have $R^{0}$ itself. Then, since $\phi(U)$ is not empty, it cannot be compact.
This idea seems right in the first place, but afterwards a confusion arose. Yes, I prove that $\phi(U)$ is not compact, but it does not contradict the homeomorphism, since $M$ is compact manifold, but $U$ is not necessarily compact. Or a compact manifold is naturally covered by a compact covering? or the covering itself is naturally compact?
Please feel free to point out where I got wrong
Thank you!
Since $U$ is a chart, then$U \subset M$, but since $U$ covers $M$, $M \subset U$. Thus, $M=U$.
Thus the chart now is actually $(M,\phi)$.
By the definition of local Euclidean, $M$ is homeomorphic to an open subset $\phi(M) \subset R^{n}$.
But the only connected open compact subsets of $R^{n}$ are empty set and $R^{0}$ if $n=0$, but $\phi(M)$ is not empty, thus $\phi(M)$ is not compact, which contradicts the homeomorphism.
Also, I appreciate the help from the comment, thank you so much!